5.1 the Natural Logarithmic Function: Differentiation

318 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

5.1 The Natural Logarithmic Function: Differentiation

Develop and use properties of the natural logarithmic function. Understand the definition of the number e. Find derivatives of functions involving the natural logarithmic function.

The Natural Logarithmic Function Recall that the General Power Rule x nϩ1 ͵x n dx ϭ ϩ C, n Ϫ1 General Power Rule n ϩ 1 has an important disclaimer—it doesn’t apply when n ϭϪ1. Consequently, you have not yet found an antiderivative for the function f ͑x͒ ϭ 1͞x. In this section, you will use the Second Fundamental Theorem of Calculus to define such a function. This antiderivative is a function that you have not encountered previously in the text. It is neither algebraic nor trigonometric, but falls into a new class of functions called logarithmic functions. This particular function is the natural logarithmic function.

JOHN NAPIER (1550–1617) Definition of the Natural Logarithmic Function Logarithms were invented by the Scottish mathematician John The natural logarithmic function is defined by Napier. Napier coined the term x 1 logarithm, from the two Greek ln x ϭ ͵ dt, x > 0. words logos (or ratio) and 1 t arithmos (or number), to describe the theory that he spent 20 years The domain of the natural logarithmic function is the set of all positive real developing and that first appeared numbers. in the book Mirifici Logarithmorum canonis descriptio (A Description of the Marvelous Rule of Logarithms). Although he did not From this definition, you can see that ln x is positive for x > 1 and negative for introduce the natural logarithmic 0 < x < 1, as shown in Figure 5.1. Moreover,ln͑1͒ ϭ 0, because the upper and lower function, it is sometimes called the limits of integration are equal when x ϭ 1. Napierian logarithm. See LarsonCalculus.com to read y y more of this biography.

4 4 y = 1 y = 1 t t 3 3 x x If x > 1, then ∫ 1 dt > 0. If 0 < x < 1, then ∫ 1 dt < 0. 2 1 t 2 1 t

1 1

x t x t 1 2 3 4 1 2 3 4 If x > 1, then ln x > 0. If 0 < x < 1, then ln x < 0. Figure 5.1

Exploration Graphing the Natural Logarithmic Function Using only the definition of the natural logarithmic function, sketch a graph of the function. Explain your reasoning.

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y To sketch the graph of y ϭ ln x, you can think of the natural logarithmic function y = ln x as an antiderivative given by the differential equation 1 (1, 0) dy 1 x ϭ . 1 2345 dx x − 1 Figure 5.2 is a computer-generated graph, called a slope field (or direction field), ͞ ϭ −2 showing small line segments of slope 1 x. The graph of y ln x is the solution that passes through the point ͑1, 0͒. (You will study slope fields in Section 6.1.) −3 The next theorem lists some basic properties of the natural logarithmic function.

1 Each small line segment has a slope of . x THEOREM 5.1 Properties of the Natural Logarithmic Function Figure 5.2 The natural logarithmic function has the following properties. 1. The domain is ͑0, ϱ͒ and the range is ͑Ϫϱ, ϱ͒. 2. The function is continuous, increasing, and one-to-one. 3. The graph is concave downward.

y y′ = 1 Proof The domain of f ͑x͒ ϭ ln x is ͑0, ϱ͒ by definition. Moreover, the function is y′ = 1 4 3 continuous because it is differentiable. It is increasing because its derivative y′ = 1 x = 4 1 2 x = 3 y = ln x 1 y′ = 1 x = 2 fЈ ͑x͒ ϭ First derivative x x y′ = 2 1 234 x = 1 is positive for x > 0, as shown in Figure 5.3. It is concave downward because −1 x = 1 y′ = 3 2 1 x = 1 fЉ ͑x͒ ϭϪ Second derivative y′ = 4 3 x2 −2 x = 1 4 is negative for x > 0. The proof that f is one-to-one is given in Appendix A. The The natural logarithmic function is increasing, and its graph is concave following limits imply that its range is the entire real number line. downward. ϭϪϱ limϩ ln x Figure 5.3 x→0 and lim ln x ϭ ϱ x→ϱ Verification of these two limits is given in Appendix A. See LarsonCalculus.com for Bruce Edwards’s video of this proof.

Using the definition of the natural logarithmic function, you can prove several important properties involving operations with natural logarithms. If you are already familiar with logarithms, you will recognize that these properties are characteristic of all logarithms.

THEOREM 5.2 Logarithmic Properties If a and b are positive numbers and n is rational, then the following properties are true. 1. ln͑1͒ ϭ 0 2. ln͑ab͒ ϭ ln a ϩ ln b 3. ln͑an͒ ϭ n ln a a 4. ln΂ ΃ ϭ ln a Ϫ ln b b

Proof The first property has already been discussed. The proof of the second property follows from the fact that two antiderivatives of the same function differ at most by a constant. From the Second Fundamental Theorem of Calculus and the definition of the natural logarithmic function, you know that d d x 1 1 ͓ln x͔ ϭ ΄͵ dt΅ ϭ . dx dx 1 t x So, consider the two derivatives d a 1 ͓ln͑ax͔͒ ϭ ϭ dx ax x and d 1 1 ͓ln a ϩ ln x͔ ϭ 0 ϩ ϭ . dx x x Because ln͑ax͒ and ͑ln a ϩ ln x͒ are both antiderivatives of 1͞x, they must differ at most by a constant. ln͑ax͒ ϭ ln a ϩ ln x ϩ C By letting x ϭ 1, you can see that C ϭ 0. The third property can be proved similarly by comparing the derivatives of ln͑xn͒ and n lnx. Finally, using the second and third properties, you can prove the fourth property. a ln΂ ΃ ϭ ln͓a͑bϪ1͔͒ ϭ ln a ϩ ln͑bϪ1͒ ϭ ln a Ϫ ln b b

See LarsonCalculus.com for Bruce Edwards’s video of this proof.

Example 1 shows how logarithmic properties can be used to expand logarithmic expressions.

Expanding Logarithmic Expressions

10 a. ln ϭ ln 10 Ϫ ln 9 Property 4 9 f(x) = ln x2 ͞ 5 b. lnΊ3x ϩ 2 ϭ ln͑3x ϩ 2͒1 2 Rewrite with rational exponent. 1 ϭ ln͑3x ϩ 2͒ Property 3 2

−5 5 6x c. ln ϭ ln͑6x͒ Ϫ ln 5 Property 4 5 ϭ ln 6 ϩ ln x Ϫ ln 5 Property 2 2 2 −5 ͑x ϩ 3͒ d. ln ϭ ln͑x 2 ϩ 3͒ 2 Ϫ ln͑xΊ3 x 2 ϩ 1 ͒ xΊ3 x 2 ϩ 1 ϭ ͑ 2 ϩ ͒ Ϫ ͓ ϩ ͑ 2 ϩ ͒1͞3͔ g(x) = 2 ln x 2 ln x 3 ln x ln x 1 5 ϭ 2 ln͑x 2 ϩ 3͒ Ϫ ln x Ϫ ln͑x 2 ϩ 1͒1͞3 1 ϭ 2 ln͑x 2 ϩ 3͒ Ϫ ln x Ϫ ln͑x 2 ϩ 1͒ 3 −5 5

When using the properties of logarithms to rewrite logarithmic functions, you must check to see whether the domain of the rewritten function is the same as the domain of −5 the original. For instance, the domain of f͑x͒ ϭ ln x 2 is all real numbers except x ϭ 0, Figure 5.4 and the domain of g͑x͒ ϭ 2 ln x is all positive real numbers. (See Figure 5.4.)

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.1 The Natural Logarithmic Function: Differentiation 321 The Number e It is likely that you have studied logarithms in an y algebra course. There, without the benefit of calculus, 3 logarithms would have been defined in terms of a y = 1 t base number. For example, common logarithms have a base of 10 and therefore log 10 ϭ 1. (You 2 10 e Area = ∫ 1 dt = 1 will learn more about this in Section 5.5.) 1 t The base for the natural logarithm is defined 1 using the fact that the natural logarithmic function

is continuous, is one-to-one, and has a range of t ͑Ϫϱ, ϱ͒. So, there must be a unique real number x 1 2 3 such that ln x ϭ 1, as shown in Figure 5.5. This e ≈ 2.72 number is denoted by the letter e. It can be shown e is the base for the natural that e is irrational and has the following decimal logarithm because ln e ϭ 1. approximation. Figure 5.5

e Ϸ 2.71828182846

Definition of e The letter e denotes the positive real number such that e1 ln e ϭ ͵ dt ϭ 1. 1 t

FOR FURTHER INFORMATION To learn more about the number e, see the article “Unexpected Occurrences of the Number e ” by Harris S. Shultz and Bill Leonard in Mathematics Magazine. To view this article, go to MathArticles.com.

Once you know that ln e ϭ 1, you can use logarithmic properties to evaluate the natural logarithms of several other numbers. For example, by using the property ln͑e n͒ ϭ n ln e ϭ n͑1͒ ϭ n y you can evaluate ln͑e n͒ for various values of n, as shown in the table and in Figure 5.6. y = ln x (e2, 2) 2 (e, 1) 1 1 1 1 0 Ϸ Ϸ Ϸ 0 2 (e , 0) x 3 0.050 2 0.135 0.368 e ϭ 1 e Ϸ 2.718 e Ϸ 7.389 x e e e 1 2 345678 − Ϫ Ϫ Ϫ −1 (e 1, −1) ln x 3 2 10 1 2 −2 −2 (e , −2) − The logarithms shown in the table above are convenient because the x- values are −3 (e 3, −3) integer powers of e. Most logarithmic expressions are, however, best evaluated with a If x ϭ en, then ln x ϭ n. calculator. Figure 5.6 Evaluating Natural Logarithmic Expressions

a. ln 2 Ϸ 0.693 b. ln 32 Ϸ 3.466 c. ln 0.1 Ϸ Ϫ2.303

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 322 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions The Derivative of the Natural Logarithmic Function The derivative of the natural logarithmic function is given in Theorem 5.3. The first part of the theorem follows from the definition of the natural logarithmic function as an antiderivative. The second part of the theorem is simply the Chain Rule version of the first part.

THEOREM 5.3 Derivative of the Natural Logarithmic Function Let u be a differentiable function of x. d 1 1. ͓ln x͔ ϭ , x > 0 dx x d 1 du uЈ 2. ͓ln u͔ ϭ ϭ , u > 0 dx u dx u

Differentiation of Logarithmic Functions

See LarsonCalculus.com for an interactive version of this type of example. d uЈ 2 1 a. ͓ln͑2x͔͒ ϭ ϭ ϭ u ϭ 2x dx u 2x x d uЈ 2x b. ͓ln͑x 2 ϩ 1͔͒ ϭ ϭ u ϭ x 2 ϩ 1 dx u x 2 ϩ 1 d d d c. ͓x ln x͔ ϭ x ΂ ͓ln x͔΃ ϩ ͑ln x͒΂ ͓x͔΃ Product Rule dx dx dx 1 ϭ x΂ ΃ ϩ ͑ln x͒͑1͒ x ϭ 1 ϩ ln x d d d. ͓͑ln x͒3͔ ϭ 3͑ln x͒2 ͓ln x͔ Chain Rule dx dx 1 ϭ 3͑ln x͒2 x

Napier used logarithmic properties to simplify calculations involving products, quotients, and powers. Of course, given the availability of calculators, there is now little need for this particular application of logarithms. However, there is great value in using logarithmic properties to simplify differentiation involving products, quotients, and powers.

Logarithmic Properties as Aids to Differentiation

Differentiate f͑x͒ ϭ lnΊx ϩ 1. Solution Because

͞ 1 f͑x͒ ϭ lnΊx ϩ 1 ϭ ln͑x ϩ 1͒1 2 ϭ ln͑x ϩ 1͒ Rewrite before differentiating. 2 you can write 1 1 1 fЈ͑x͒ ϭ ΂ ΃ ϭ . Differentiate. 2 x ϩ 1 2͑x ϩ 1͒

Logarithmic Properties as Aids to Differentiation

Differentiate x͑x2 ϩ 1͒2 f͑x͒ ϭ ln . Ί2x3 Ϫ 1 Solution Because x͑x2 ϩ 1͒2 f ͑x͒ ϭ ln Write original function. Ί2x3 Ϫ 1 1 ϭ ln x ϩ 2 ln͑x2 ϩ 1͒ Ϫ ln͑2x3 Ϫ 1͒ Rewrite before differentiating. 2 you can write 1 2x 1 6x2 fЈ͑x͒ ϭ ϩ 2΂ ΃ Ϫ ΂ ΃ Differentiate. x x2 ϩ 1 2 2x3 Ϫ 1 1 4x 3x2 ϭ ϩ Ϫ . Simplify. x x2 ϩ 1 2x3 Ϫ 1

In Examples 4 and 5, be sure you see the benefit of applying logarithmic properties before differentiating. Consider, for instance, the difficulty of direct differentiation of the function given in Example 5. On occasion, it is convenient to use logarithms as aids in differentiating nonlogarithmic functions. This procedure is called logarithmic differentiation.

Logarithmic Differentiation

Find the derivative of ͑x Ϫ 2͒2 y ϭ , x 2. Ίx 2 ϩ 1 Solution Note that y > 0 for all x 2. So,ln y is defined. Begin by taking the natural logarithm of each side of the equation. Then apply logarithmic properties and differentiate implicitly. Finally, solve for yЈ. ͑x Ϫ 2͒2 y ϭ , x 2 Write original equation. Ίx2 ϩ 1 ͑x Ϫ 2͒ 2 ln y ϭ ln Take natural log of each side. Ίx 2 ϩ 1 1 ln y ϭ 2 ln͑x Ϫ 2͒ Ϫ ln͑x 2 ϩ 1͒ Logarithmic properties 2 yЈ 1 1 2x ϭ 2΂ ΃ Ϫ ΂ ΃ Differentiate. y x Ϫ 2 2 x 2 ϩ 1 yЈ x2 ϩ 2x ϩ 2 ϭ Simplify. y ͑x Ϫ 2͒͑x2 ϩ 1͒ x2 ϩ 2x ϩ 2 yЈ ϭ y΄ ΅ Solve for yЈ. ͑x Ϫ 2͒͑x2 ϩ 1͒ ͑x Ϫ 2͒2 x2 ϩ 2x ϩ 2 yЈ ϭ ΄ ΅ Substitute for y. Ίx2 ϩ 1 ͑x Ϫ 2͒͑x2 ϩ 1͒ ͑x Ϫ 2͒͑x2 ϩ 2x ϩ 2͒ yЈ ϭ Simplify. ͑x 2 ϩ 1͒3͞ 2

Because the natural logarithm is undefined for negative numbers, you will often encounter expressions of the form lnԽuԽ. The next theorem states that you can differentiate functions of the form y ϭ lnԽuԽ as though the absolute value notation was not present.

THEOREM 5.4 Derivative Involving Absolute Value If u is a differentiable function of x such that u 0, then d uЈ ͓lnԽuԽ͔ ϭ . dx u

Proof If u > 0, then ԽuԽ ϭ u, and the result follows from Theorem 5.3. If u < 0, then ԽuԽ ϭϪu, and you have d d ͓lnԽuԽ͔ ϭ ͓ln͑Ϫu͔͒ dx dx ϪuЈ ϭ Ϫu uЈ ϭ . u

See LarsonCalculus.com for Bruce Edwards’s video of this proof.

Derivative Involving Absolute Value

Find the derivative of f͑x͒ ϭ lnԽcos xԽ. Solution Using Theorem 5.4, let u ϭ cos x and write

d uЈ d uЈ ͓lnԽcos xԽ͔ ϭ ͓lnԽuԽ͔ ϭ dx u dx u Ϫsin x ϭ u ϭ cos x cos x ϭϪtan x. Simplify.

Finding Relative Extrema

Locate the relative extrema of y y ϭ ln͑x 2 ϩ 2x ϩ 3͒. 2 Solution Differentiating y, you obtain y = ln(x2 + 2x + 3) ϩ dy ϭ 2x 2 . dx x 2 ϩ 2x ϩ 3 (−1, ln 2) Because dy͞dx ϭ 0 when x ϭϪ1, you can apply the First Derivative Test and Relative minimum conclude that the point ͑Ϫ1, ln 2͒ is a x relative minimum. Because there are −2 −1 no other critical points, it follows that this is the only relative extremum. The derivative of y changes from (See Figure 5.7.) negative to positive at x ϭϪ1. Figure 5.7

5.1 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Evaluating a Logarithm In Exercises 1–4, use a graphing x Ϫ 1 25.lnΊ 26. ln͑3e2͒ utility to evaluate the logarithm by (a) using the natural x logarithm key and (b) using the integration capabilities to ͐xͧ ͨ 1 evaluate the integral 1 1/t dt. 27.ln z͑z Ϫ 1͒2 28. ln e 1. ln 45 2. ln 8.3 3. ln 0.8 4. ln 0.6 Condensing a Logarithmic Expression In Exercises 29–34, write the expression as a logarithm of a single quantity.

Matching In Exercises 5–8, match the function with its 29. ln͑x Ϫ 2͒ Ϫ ln͑x ϩ 2͒ graph. [The graphs are labeled (a), (b), (c), and (d).] 30. 3 ln x ϩ 2 ln y Ϫ 4 ln z y y (a) (b) 1͓ ͑ ϩ ͒ ϩ Ϫ ͑ 2 Ϫ ͔͒ 31. 3 2 ln x 3 ln x ln x 1 2 4 32. 2͓ln x Ϫ ln͑x ϩ 1͒ Ϫ ln͑x Ϫ 1͔͒ 1 3 Ϫ 1 ͑ 2 ϩ ͒ x 2 33. 2 ln 3 2 ln x 1 2 3 4 5 −1 3͓ ͑ 2 ϩ ͒ Ϫ ͑ ϩ ͒ Ϫ ͑ Ϫ ͔͒ 1 34. 2 ln x 1 ln x 1 ln x 1 −2 x 12345 −3 Verifying Properties of Logarithms In Exercises 35 and g by using a graphing utility to graph f ؍ a) verify that f) ,36 g ؍ c)y (d) y and g in the same viewing window and (b) verify that f) algebraically. 2 2 1 x 2 35. f͑x͒ ϭ ln , x > 0, g͑x͒ ϭ 2 ln x Ϫ ln 4 x 4 x 143 5 −4 −3 −1 −1 1 − 36. f͑x͒ ϭ lnΊx͑x 2 ϩ 1͒, g͑x͒ ϭ ͓ln x ϩ ln͑x 2 ϩ 1͔͒ 1 −2 2 − − 2 3 Finding a Limit In Exercises 37–40, find the limit.

5.f͑x͒ ϭ ln x ϩ 1 6. f͑x͒ ϭϪln x 37.lim ln͑x Ϫ 3͒ 38. lim ln͑6 Ϫ x͒ x→3ϩ x→6Ϫ 7.f͑x͒ ϭ ln͑x Ϫ 1͒ 8. f͑x͒ ϭϪln͑Ϫx͒ x 39.lim ln͓x 2͑3 Ϫ x͔͒ 40. lim ln x→2Ϫ x→5ϩ Ίx Ϫ 4 Sketching a Graph In Exercises 9–16, sketch the graph of the function and state its domain. Finding a Derivative In Exercises 41–64, find the derivative of the function. 9.f͑x͒ ϭ 3 ln x 10. f͑x͒ ϭϪ2 ln x ͑ ͒ ϭ ͑ ͒ ͑ ͒ ϭ ͑ Ϫ ͒ 11.f͑x͒ ϭ ln 2x 12. f͑x͒ ϭ lnԽxԽ 41.f x ln 3x 42. f x ln x 1 ͑ ͒ ϭ 2 ͑ ͒ ϭ ͑ 2 ϩ ͒ 13.f ͑x͒ ϭ ln ͑x Ϫ 3͒ 14. f ͑x͒ ϭ ln x Ϫ 4 43.g x ln x 44. h x ln 2x 1 ϭ ͑ ͒4 ϭ 2 15.h͑x) ϭ ln͑x ϩ 2) 16. f͑x͒ ϭ ln͑x Ϫ 2) ϩ 1 45.y ln x 46. y x ln x 47.y ϭ ln͑t ϩ 1͒2 48. y ϭ lnΊx2 Ϫ 4 Using Properties of Logarithms In Exercises 17 and 18, 49.y ϭ ln͑xΊx2 Ϫ 1 ͒ 50. y ϭ ln͓t͑t2 ϩ 3͒3] use the properties of logarithms to approximate the indicated logarithms, given that ln 2 y 0.6931 and ln 3 y 1.0986. x 2x 51.f͑x͒ ϭ ln΂ ΃ 52. f͑x͒ ϭ ln΂ ΃ x2 ϩ 1 x ϩ 3 17. (a)ln 6 (b)ln 2 (c)ln 81 (d) lnΊ 3 3 ln t ln t Ί3 1 53.g͑t͒ ϭ 54. h͑t͒ ϭ 18. (a)ln 0.25 (b)ln 24 (c)ln 12 (d) ln 72 t 2 t ϭ ͑ 2͒ ϭ ͑ ͒ Expanding a Logarithmic Expression In Exercises 55.y ln ln x 56. y ln ln x 19–28, use the properties of logarithms to expand the logarith- x ϩ 1 x Ϫ 1 57.y ϭ lnΊ 58. y ϭ lnΊ3 mic expression. x Ϫ 1 x ϩ 1 x Ί4 ϩ x2 19.ln 20. lnΊx5 59.f͑x͒ ϭ ln΂ ΃ 60. f͑x͒ ϭ ln͑x ϩ Ί4 ϩ x 2 ͒ 4 x xy 61.y ϭ lnԽsin xԽ 62. y ϭ lnԽcsc xԽ 21.ln 22. ln͑xyz͒ z cos x 63. y ϭ ln 64. y ϭ lnԽsec x ϩ tan xԽ 23.ln͑xΊx2 ϩ 5͒ 24. lnΊa Ϫ 1 Խcos x Ϫ 1Խ

Finding an Equation of a Tangent Line In Exercises Logarithmic Differentiation In Exercises 89–94, use 65–72, (a) find an equation of the tangent line to the graph of f logarithmic differentiation to find dy/dx. at the given point, (b) use a graphing utility to graph the ϭ Ί 2 ϩ function and its tangent line at the point, and (c) use the 89. y x x 1, x > 0 derivative feature of a graphing utility to confirm your results. 90. y ϭ Ίx2͑x ϩ 1͒͑x ϩ 2͒, x > 0 2Ί ϭ 4 ͑ ͒ x 3x Ϫ 2 2 65. y ln x , 1, 0 91. y ϭ , x > ͑x ϩ 1͒2 3 66. y ϭ ln x3/2, ͑1, 0͒ x2 Ϫ 1 67. f͑x͒ ϭ 3x2 Ϫ ln x, ͑1, 3͒ 92. y ϭ Ί , x > 1 x2 ϩ 1 68. f͑x͒ ϭ 4 Ϫ x2 Ϫ ln͑1 x ϩ 1͒, ͑0, 4͒ 2 x͑x Ϫ 1͒3͞2 ␲ 3 93. y ϭ , x > 1 69. f ͑x͒ ϭ lnΊ1 ϩ sin2 x, ΂ , ln ΃ Ίx ϩ 1 4 Ί2 ͑x ϩ 1͒͑x Ϫ 2͒ ͑ ͒ ϭ 2 ͑ ͒ 94. y ϭ , x > 2 70. f x sin 2x ln x , 1, 0 ͑x Ϫ 1͒͑x ϩ 2͒ 71. f ͑x͒ ϭ x3 ln x, ͑1, 0͒ 1 WRITING ABOUT CONCEPTS 72. f ͑x͒ ϭ x ln x2, ͑Ϫ1, 0͒ 2 95. Properties In your own words, state the properties of the natural logarithmic function. Finding a Derivative Implicitly In Exercises 73–76, use implicit differentiation to find dy/dx. 96. Base Define the base for the natural logarithmic function. 97. Comparing Functions Let f be a function that is 2 Ϫ ϩ 2 ϭ ϩ ϭ 73.x 3 ln y y 10 74. ln xy 5x 30 positive and differentiable on the entire real number line. 75.4x3 ϩ ln y2 ϩ 2y ϭ 2x 76. 4xy ϩ ln x2y ϭ 7 Let g͑x͒ ϭ ln f͑x͒. (a) When g is increasing, must f be increasing? Explain. Differential Equation In Exercises 77 and 78, show that the function is a solution of the differential equation. (b) When the graph of f is concave upward, must the graph of g be concave upward? Explain. Function Differential Equation 77. y ϭ 2 ln x ϩ 3 xyЉ ϩ yЈ ϭ 0 78. y ϭ x ln x Ϫ 4x x ϩ y Ϫ xyЈ ϭ 0 98. HOW DO YOU SEE IT? The graph shows the ͑ Њ ͒ Relative Extrema and Points of Inflection In Exercises temperature T in C of an object h hours after it 79–84, locate any relative extrema and points of inflection. Use is removed from a furnace. a graphing utility to confirm your results. T x2 160 79.y ϭ Ϫ ln x 80. y ϭ 2x Ϫ ln͑2x͒

C) 140

2 ° 120 ln x 81.y ϭ x ln x 82. y ϭ 100 x 80 x x 60 83.y ϭ 84. y ϭ x2 ln ln x 4 40 Temperature (in 20 h Linear and Quadratic Approximation In Exercises 85 12345678 and 86, use a graphing utility to graph the function. Then graph Hours ͨ Јͧ ͨͧ ؊ ͨ ͧ ؍ ͨ ͧ P1 x f 1 1 f 1 x 1 (a) Find lim T. What does this limit represent? h→ϱ and (b) When is the temperature changing most rapidly?

Јͧ ͨͧ ؊ ͨ 1 Љͧ ͨͧ ؊ ͒2 ͨ ͧ ؍ ͨ ͧ P2 x f 1 1 f 1 x 1 1 2 f 1 x 1 in the same viewing window. Compare the values of f, P1, P2, True or False? In Exercises 99–102, determine whether the ؍ and their first derivatives at x 1. statement is true or false. If it is false, explain why or give an 85.f͑x͒ ϭ ln x 86. f͑x͒ ϭ x ln x example that shows it is false. 99. ln͑x ϩ 25͒ ϭ ln x ϩ ln 25 Using Newton’s Method In Exercises 87 and 88, use 100. ln xy ϭ ln x ln y Newton’s Method to approximate, to three decimal places, the x-coordinate of the point of intersection of the graphs of the 101. If y ϭ ln ␲, then yЈ ϭ 1͞␲. two equations. Use a graphing utility to verify your result. 102. If y ϭ ln e, then yЈ ϭ 1. 87.y ϭ ln x, y ϭϪx 88. y ϭ ln x, y ϭ 3 Ϫ x

103. Home Mortgage The term t (in years) of a $200,000 106. Modeling Data The atmospheric pressure decreases home mortgage at 7.5% interest can be approximated by with increasing altitude. At sea level, the average air pressure is one atmosphere (1.033227 kilograms per square centimeter). ϭ x The table shows the pressures p (in atmospheres) at selected t 13.375 ln΂ Ϫ ΃, x > 1250 x 1250 altitudes h (in kilometers). where x is the monthly payment in dollars. h 0 5 10 15 20 25 (a) Use a graphing utility to graph the model. (b) Use the model to approximate the term of a home p 1 0.55 0.25 0.12 0.06 0.02 mortgage for which the monthly payment is $1398.43. What is the total amount paid? (a) Use a graphing utility to find a model of the form ϭ ϩ (c) Use the model to approximate the term of a home p a b ln h for the data. Explain why the result is an mortgage for which the monthly payment is $1611.19. error message. What is the total amount paid? (b) Use a graphing utility to find the logarithmic model ϭ ϩ (d) Find the instantaneous rates of change of t with respect to h a b ln p for the data. x when x ϭ $1398.43 and x ϭ $1611.19. (c) Use a graphing utility to plot the data and graph the (e) Write a short paragraph describing the benefit of the model. higher monthly payment. (d) Use the model to estimate the altitude when p ϭ 0.75. ϭ 104. Sound Intensity (e) Use the model to estimate the pressure when h 13. The relationship between (f) Use the model to find the rates of change of pressure ϭ ϭ the number of decibels when h 5 and h 20. Interpret the results. ␤ and the intensity of 107. Tractrix A person walking along a dock drags a boat by a a sound I in watts per 10-meter rope. The boat travels along a path known as a centimeter squared is tractrix (see figure). The equation of this path is

I 10 ϩ Ί100 Ϫ x2 ␤ ϭ 10 ϭ Ϫ Ί Ϫ 2 ln ΂ Ϫ ΃. y 10 ln΂ ΃ 100 x . ln 10 10 16 x

(a) Use the properties of (a) Use a graphing utility to y logarithms to write the formula in simpler form. graph the function. 10 (b) Determine the number of decibels of a sound with an (b) What are the slopes of Tractrix intensity of 10Ϫ5 watt per square centimeter. this path when x ϭ 5 and ϭ x 9? 5 (c) What does the slope of 105. Modeling Data The table shows the temperatures T (in the path approach as Њ x F) at which water boils at selected pressures p (in pounds per x → 10? 5 10 square inch). (Source: Standard Handbook of Mechanical Engineers) 108. Prime Number Theorem There are 25 prime numbers less than 100. The Prime Number Theorem states that the number of primes less than x approaches p 5 10 14.696 (1 atm) 20 x T 162.24Њ 193.21Њ 212.00Њ 227.96Њ p͑x͒ Ϸ . ln x p 30 40 60 80 100 Use this approximation to estimate the rate (in primes per 100 integers) at which the prime numbers occur when T 250.33Њ 267.25Њ 292.71Њ 312.03Њ 327.81Њ (a) x ϭ 1000. A model that approximates the data is (b) x ϭ 1,000,000. (c) x ϭ 1,000,000,000. T ϭ 87.97 ϩ 34.96 ln p ϩ 7.91Ίp. 109. Conjecture Use a graphing utility to graph f and g in the (a) Use a graphing utility to plot the data and graph the same viewing window and determine which is increasing at model. the greater rate for large values of x. What can you conclude (b) Find the rates of change of T with respect to p when about the rate of growth of the natural logarithmic function? p ϭ 10 and p ϭ 70. (a) f ͑x͒ ϭ ln x, g͑x͒ ϭ Ίx (c) Use a graphing utility to graph TЈ. Find lim TЈ͑p͒ and ͑ ͒ ϭ ͑ ͒ ϭ Ί4 p→ϱ (b) f x ln x, g x x interpret the result in the context of the problem. Christopher Dodge/Shutterstock.com

1 4 Ί 1 67. 4 sin x C 69. 2 1 sin C 2 ͵ Ί 4 Ί 9. Proof 11.3 13. 1 1 x dx 2 1 3 0 71. ͑1 sec x͒ C v 3 15. (a) 100 1͑ 2͒3͞2 73. (a) Answers will vary. (b) y 3 9 x 5 80 Sample answer: 3 60 y −6 6 40 2 20

x t 0.2 0.4 0.6 0.8 1.0 −33 −5 (b)͑0, 0.4͒ and ͑0.7, 1.0͒ (c) 150 mi͞h2 (d) Total distance traveled in miles; 38.5 mi (e) Sample answer: 100 mi͞h2 17. (a)–(c) Proofs 455 ͞ 468 75.2 77. 2 79. 28 15 81. 2 83. 7 ͑ ͒ ͑ ͒ ͑ ͒ 64 32 96 19. (a) R n , I, T n , L n 85. (a) (b) (c) (d) 32 1 5 5 5 (b) S͑4͒ ͓f ͑0͒ 4f ͑1͒ 2f ͑2͒ 4f ͑3͒ f ͑4͔͒ Ϸ 5.42 87. Trapezoidal Rule: 0.285 89. Trapezoidal Rule: 0.637 3 Simpson’s Rule: 0.284 Simpson’s Rule: 0.685 Chapter 5 Graphing Utility: 0.284 Graphing Utility: 0.704 Section 5.1 (page 325) P.S. Problem Solving (page 315) 45 1 ͑ ͒ ͑ ͒ ͞ ͑ ͒ 1. (a) 3.8067 (b) ln 45 ͵ dt Ϸ 3.8067 1. (a) L 1 0 (b) L x 1 x, L 1 1 t Ϸ 1 (c) x 2.718 (d) Proof 0.8 1 n n n 3. (a) 0.2231 (b) ln 0.8 ͵ dt Ϸ 0.2231 32 4 64 3 32 2 t 3. (a) lim ΄ ͚i ͚i ͚i ΅ 1 n→ 5 4 3 n i1 n i1 n i1 5. b 6. d 7. a 8. c (b) ͑16n4 16͒͑͞15n4͒ (c) 16͞15 9. y 11. y y 5. (a) 3 2 2 2 1 1 1 x 132 54 x −1 x 13 321 −2 − 1 −1 −3 −2 Domain: x > 0 Domain: x > 0 y y (b) y 13. 15. 4 1.00 3 3 2 0.75 2 1 1 0.50 x x − − − 123 4567 32 123 0.25 1 −1 −2 x −2 −3 1232 3 5 6 7 22 − −0.25 −4 3

(c) Relative maxima at x Ί2, Ί6 Domain: x > 3 Domain: x > 2 Relative minima at x 2, 2Ί2 17. (a) 1.7917 (b) 0.4055 (c) 4.3944 (d) 0.5493 (d) Points of inflection at x 1, Ί3, Ί5, Ί7 19. ln x ln 4 21. ln x ln y ln z 1 1 y ͑ 2 ͒ ͓ ͑ ͒ ͔ 7. (a) 23. ln x 2 ln x 5 25. 2 ln x 1 ln x 5 ͑ ͒ x 2 4 27. ln z 2 ln z 1 29. ln 3 (8, 3) x 2 (6, 2) 2 f x͑x 3͒2 9 1 (0, 0) 31. ln Ί3 33. ln x x2 1 Ίx2 1 −1 24 56789 2 −2 x (2, −2) 3 2 −3 35. (a) (b) f͑x͒ ln ln x ln 4 − 4 4 f = g −5 2 ln x ln 4 0 9 ͑ ͒ (b) g x x 012345678

− F͑x͒ 0 1 2 7 4 7 2 1 3 3 2 2 2 4 37. 39. ln 4 Ϸ 1.3863 41.1͞x 43. 2͞x (c) x 4, 8 (d) x 2 Answers to Odd-Numbered Exercises A43

2x2 1 103. (a) 50 (b) 30 yr; $503,434.80 45. 4͑ln x͒3͞x 47. 2͑͞t 1͒ 49. x͑x2 1͒ (c) 20 yr; $386,685.60 1 x2 1 2 ln t 2 1 51. 53. 55. x͑x2 1͒ t3 x ln x2 x ln x 1 4 1000 3000 57. 59. 61. cot x 0 1 x2 x͑x2 4͒ ͞ Ϸ sin x (d) When x 1398.43, dt dx 0.0805. When 63. tan x ͞ Ϸ cos x 1 x 1611.19, dt dx 0.0287. 65. (a) y 4x 4 67. (a) 5x y 2 0 (e) Two benefits of a higher monthly payment are a shorter term and a lower total amount paid. (b) 5 (b) 4 350 30 (1, 3) 105. (a) (c)

−5 5 (1, 0) −12

− − 5 3 0 100 0 100 0 0 69. (a) y 1 x 1 1 ln͑3͒ 3 12 2 2 ͑ ͒ Ϸ ͞ ͞ 2 ͑ ͒ 2 (b) T 10 4.75 lb in. lim T p 0 (b) p → π 3 , ln 2 ( 24 ( T͑70͒ Ϸ 0.97͞lb͞in. Answers will vary. −2 2 107. (a) 20 (b) When x 5, dy͞dx Ί3. When x 9, −2 dy͞dx Ί19͞9. 71. (a) y x 1 dy 2 0 10 (b) 0 (c) lim 0 x→10 dx 109. (a) 25 (b) 15 −1 3 (1, 0) g g f −2 f ͑ 2͒ 2xy y 1 6x 0 500 0 20,000 73. 75. 0 3 2y2 1 y 0 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 77. xy y x͑2͞x2͒ ͑2͞x͒ 0 For x > 4, g x > f x . For x > 256,g x > f x . ͑ 1͒ g is increasing at a fasterg is increasing at a faster 79. Relative minimum: 1, 2 81. Relative minimum: ͑e1, e1͒ rate than f for large values rate than f for large values 83. Relative minimum:͑e, e͒; Point of inflection: ͑e2, e2͞2͒ of x. of x. ͑ ͒ ͑ ͒ ͑ ͒ 1͑ ͒2 f x ln x increases very slowly for large values of x. 85. P1 x x 1; P2 x x 1 2 x 1 2

P1 f The values of f, P1, and P2 and −15their first derivatives agree at P2 x 1.

−2 87. x Ϸ 0.567 89. ͑2x2 1͒͞Ίx2 1 3x3 15x2 8x ͑2x2 2x 1͒Ίx 1 91. 93. 2͑x 1͒3Ί3x 2 ͑x 1͒3͞2 95. The domain of the natural logarithmic function is͑0, ͒, and the range is ͑, ͒. The function is continuous, increasing, and one-to-one, and its graph is concave downward. In addition, if and ba are positive numbers and n is rational, then ln͑1͒ 0, ln͑a b͒ ln a ln b, ln͑an͒ n ln a, and ln͑a͞b͒ ln a ln b. 97. (a) Yes. If the graph of g is increasing, then g͑x͒ > 0. Because f͑x͒ > 0, you know that f͑x͒ g͑x͒f͑x͒ and thus f͑x͒ > 0. Therefore, the graph of f is increasing. (b) No. Let f͑x͒ x2 1 (positive and concave up), and let g͑x͒ ln͑x2 1͒ (not concave up). 99. False. ln x ln 25 ln 25x d 101. False. is a constant, so ͓ln ͔ 0. dx